It's certainly plausible.
However, there are lots of various technical difficulties that one would face in performing this operation, Nothing major enough to suggest it couldn't be done, but I'll describe some of them here.
The first obstacle is that most of the time chromosomes don't look like the classical pictures of rods or crosses in the nucleus. These structures only form when the chromosomes condense during mitosis or meiosis. Most of the time the chromosomes are actually loose and intermingled like a big bowl of spaghetti. What this means is that if you want to separate one chromosome from the rest you likely need to take them while they are in their portable, condensed form. Easy enough right?
But another issue is that the chromosomes of eggs and particularly sperm are heavily modified compared to normal chromosomes. Perhaps you've heard the term epigenetics which in this case is referring to the modifications of the DNA and its associated proteins. These epigenetic features are essential for properly regulating the functions of the cell. If you think about it, a liver cell and an egg cell have exactly the same chromosomes, but must do completely different things. This is accomplished to a large extent by epigenetic regulation. What this means though, is that if you take a chromosome out of a liver cell and put it into an egg cell its epigenetic state comes with it and it's not going to behave exactly like an egg cell. Specifically, sperm chromosomes are very different from other cells in that the histones that the DNA is wrapped around are mostly replaced with a different protein called a protamine. Sperm chromosomes also have very different methylation patterns.
Another issue related to the epigenetics of the chromosomes is genomic imprinting. It turns out that some genes are epigenetically programmed during the production of gametes so that they will be turned on in the sperm, but off in the egg or vice versa. Essentially, certain genes will only be expressed from your maternal genome and certain others will only be expressed from the paternal genome and this is caused by different epigenetic programs on the different chromosomes. This is critically important as dysregulation of these genes causes known disorders. So, you’ll also need to ensure that these imprinted genes are properly programmed.
So, to solve all these epigenetics problems why can't we just take chromosomes from sperm and eggs to make our chimeric sperm and eggs? Well, most of these epigenetic changes occur after the last division of that cell. This means we will never naturally encounter a separable, condensed chromosome with all of the epigenetic programming we need. There are multiple potential solutions to this problem but I think the most likely is to develop a process to reprogram chimeric cells into gametes, but this is not trivial as those processes are generally more complex than the differentiation of other cells.